[/quote]
,
such that:
;
.
and 
such that 
.
are positive isn't necessary.
such that 
is a perfect square.
and 
be non-zero integers such that 
Prove that ![\[(a_1^2-1)(a_2^2-1)\cdots(a_n^2-1)\]](http://latex.artofproblemsolving.com/b/f/f/bffd77e507fc4ddc8368c8239c523a4df40448c4.png)
is a perfect square.
be positive real numbers.Prove that![$$\sqrt[3]{ab}+\sqrt[3]{cd}\leq\sqrt[3]{(a+b+c)(c+d+a)}.$$](http://latex.artofproblemsolving.com/b/7/9/b796d0f9a2d9430bf5efb83b901427b13386f478.png)
When equality holds?
be an integer. Ion draws a regular 
safisty: 
and height 
:
Volume of cone frustum with base radii 
,
Y by
Problem 1:
A regular hexagon of side length
is rotated 
degrees about one side. The space through which the hexagon travels during the rotation forms a solid. Find the volume of this solid.
Problem 2:
A regular octagon of side length is rotated degrees about one side. The space through which the octagon travels through during the rotation forms a solid. Find the volume of this solid.
Source:Own
A regular hexagon of side length
is rotated 
degrees about one side. The space through which the hexagon travels during the rotation forms a solid. Find the volume of this solid.Problem 2:
A regular octagon of side length
is rotated degrees about one side. The space through which the octagon travels through during the rotation forms a solid. Find the volume of this solid.Source:Own
This post has been edited 11 times. Last edited by ReticulatedPython, Apr 21, 2025, 3:33 PM
and base radii 
and 
minus two cones with height 
Adding the volumes of these parts together yields 
